In IQ based signal processing, a baseband signal z(t) is represented as a complex number z(t)=zI(t)+jzQ(t) with the real part zI(t) being referred to as an in-phase (I) signal and the imaginary part zQ(t) being referred to as a quadrature phase (Q) signal. A variety of wireless communication protocols depend on IQ-based signal processing, in which the baseband signal z(t) is modulated with a carrier frequency and transmitted wirelessly.
In direct down conversion receivers or low-intermediate frequency (low-IF) receivers, system imperfections adversely affect the accuracy of the recovered I and Q signals in the analog domain, which need be estimated and compensated in the digital domain using advanced signal processing algorithms. Some of the system imperfections are caused by an imbalance between components in the I and Q paths. For example, a local oscillator (LO) of a down converter generates the frequency-independent mismatch, while amplifiers and analog-to-digital converters (ADCs) along the I/Q analog paths generate frequency-dependent mismatches. There have been algorithms to estimate and compensate for the frequency-independent mismatches caused by LO. However, existing algorithms for frequency-dependent mismatches can not achieve satisfactory performance required for practical applications. Moreover, in practice, due to the change of temperature and other environmental factors, the I/O mismatch is not constant and thus should be tracked by the compensation algorithm.
The mismatches are difficult to estimate and compensate for the follow reasons:                1. The frequency-independent phase mismatch caused by the LO must be treated separately from the frequency-dependent phase mismatch caused by the analog baseband channel because they have to be compensated differently.        2. Conventional methods to estimate frequency-independent mismatch may produce invalid results due to the existence of frequency-dependent mismatch. It is difficult to estimate two types of mismatches independently without accounting for both types jointly during the estimation.        3. The imbalance may vary with changes in the environmental factors, such as temperature.        4. The estimation need to be performed using run-time signals in receivers, for real-time tracking and must converge to valid solution quickly for tracking purpose.        
Thus, there is a need to calculate both frequency-dependent and frequency-independent I/Q mismatches jointly and efficiently, and to compensate and correct for the mismatches with high performance and in real time.